Set systems with cross LL-intersection and kk-wise LL-intersecting families

We prove some results involving cross LL-intersections of two families of subsets of [n]={1,2,…,n}[n]={1,2,…,n}. As a consequence, we derive the following results: (1) Let L={l1,l2,…,ls}L={l1,l2,…,ls} be a set of ss positive integers. If F={F1,F2,…,Fm}F={F1,F2,…,Fm} is a family of subsets of X=[n]X=[n] satisfying |Fi−Fj|∈L|Fi−Fj|∈L for i≠ji≠j, then m≤∑i=0sn−1i. (2) Let pp be a prime, k≥2k≥2, and L={l1,l2,…,ls}L={l1,l2,…,ls} and K={k1,k2,…,kr}K={k1,k2,…,kr} be two disjoint subsets of {0,1,…,p−1}{0,1,…,p−1}. Suppose FF is a family of subsets of [n][n] such that |Fi|(modp)∈K for all Fi∈FFi∈F and |F1∩⋯∩Fk|(modp)∈L for any collection of kk distinct sets from FF. If n>(r+1)(s−2r+2)n>(r+1)(s−2r+2), then |F|≤(k−1)∑i=s−2r+1sn−1i. The first result improves a result of Frankl about families with given difference sizes between subsets and the second result gives an improvement to a theorem by Grolmusz–Sudakov and a theorem by W. Cao, K.W. Hwang, and D.B. West.